幫忙解決大專數學 - Ans.fyi 參考答案

幫忙解決大專數學

[匿名]

2012-11-11 6:43 am

可唔可以幫下手prove 一題Strong form of induction 關於Fundamental Theorem of Arithmetic

"Prove that every integer n >=2 can be written as a product of one or more (not necessarily distinct) primes, that is, n=p1p2.......pr for some prime numbers p1, p2, ......, pr"

(Hint: In the inductive step, if k+1 is prime, then we are done. Otherwise, you may need to use the following definition of composite numbers: An integer n>=2 is called composite if it can be factored as n=ab where a, b are integers satisfying 2<=a,b<n.)

回答 (1)

用戶9112

2012-11-11 7:08 am

✔ 最佳答案

Since 2 is a prime so for n=2 the statement is trueSuppose the statement is true for n≤kThen when n=k+1, case (I):k+1 is prime, then the proof is done.case (II):k+1 is composite then k+1=ab where a and b are positive integers and 2<=a, b<k+1Since the statement is true for n≤k so a and b can both be expressed as products of primesIt follows that k+1=ab can also be expressed as products of primesThus the statement is proved inductively

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